DE. PLUCKER ON A NEW GEOMETRY OE SPACE. 
747 
and by putting 
is reduced to 
x°= — 
B" 
w 
C" 
(sg—r<r)= — 'jTf=k. 
43. By successive substitution we obtain 
& — — j?ii 
C'F'+A'D' 
— ~ D ,2 + F ' 2 
CF + (AD - BE) (D 2 + E 9 ) 
~ (D 2 + E 2 ) 2 ^ + F 2 
and finally, on observing that 
2 B 2 
COS CC — J) 2 _J_ JJ 2 ’ 
the symmetrical expression 
AD-BE + CF 
fC—— J)2 + E 2 + p2 • 
. . (50) 
. . (51) 
(52) 
In order to replace OZ and OX by each other, we may make use of the formulae (41) 
and (42) on putting a=^r. By means of the last of these formulae the equation of the 
linear complex (51) is immediately transformed into the following one, 
t r=Jcr • (53) 
the constant h being the same as before. 
Again, on interchanging OY and OX we get 
%=ks (54) 
44. If Jc become equal to zero the complex is of a peculiar description, all its rays 
meet a fixed line. If the complex be represented by the general equation 
A/-f-Bs-(-C-j-D(7-{ - Eg-|-F(5g | — r<r)=0, . . . (7) 
this peculiar case is indicated by the following condition, 
AD-BE+CF=0 (55) 
45. By eliminating from the general equation of the complex or, § and (s§ — r< t) by 
means of the equations 
cc—rz- bg, 
y— sz-\- or, 
sx—ry—s^—ra. 
we get 
(A-fE^~E^+(B+F^-D^>+(CH-D^+E^)=:0. 
